If `|z_(1)-1|lt 1,|z_(2)-2|lt 2,|z_(3)-3|lt 3`, then `|z_(1)+z_(2)+z_(3)|`

Options

A. is less than 6
B. is more than 3
C. is less than 12
D. lies between 6 and 12

Answer

To solve the problem, we need to analyze the given inequalities involving the complex numbers \( z_1, z_2, \) and \( z_3 \).

### Step-by-Step Solution:

1. **Understanding the given inequalities:**
We have the following inequalities:
\[ |z_1 - 1| < 1, \quad |z_2 - 2| < 2, \quad |z_3 - 3| < 3 \]

2. **Interpreting the inequalities:**
Each inequality describes a circle in the complex plane:
- For \( |z_1 - 1| < 1 \): This means \( z_1 \) lies within a circle centered at \( 1 \) with a radius of \( 1 \). Therefore, \( z_1 \) can take values in the range:
\[ 1 - 1 < z_1 < 1 + 1 \implies 0 < z_1 < 2 \]

- For \( |z_2 - 2| < 2 \): This means \( z_2 \) lies within a circle centered at \( 2 \) with a radius of \( 2 \). Therefore, \( z_2 \) can take values in the range:
\[ 2 - 2 < z_2 < 2 + 2 \implies 0 < z_2 < 4 \]

- For \( |z_3 - 3| < 3 \): This means \( z_3 \) lies within a circle centered at \( 3 \) with a radius of \( 3 \). Therefore, \( z_3 \) can take values in the range:
\[ 3 - 3 < z_3 < 3 + 3 \implies 0 < z_3 < 6 \]

3. **Adding the inequalities:**
Now, we add the ranges for \( z_1, z_2, \) and \( z_3 \):
- The minimum value of \( z_1 + z_2 + z_3 \):
\[ 0 + 0 + 0 = 0 \]
- The maximum value of \( z_1 + z_2 + z_3 \):
\[ 2 + 4 + 6 = 12 \]

4. **Conclusion about the modulus:**
Therefore, we can conclude that:
\[ 0 < z_1 + z_2 + z_3 < 12 \]
Taking the modulus, we have:
\[ |z_1 + z_2 + z_3| < 12 \]

### Final Answer:
Thus, the final result is:
\[ |z_1 + z_2 + z_3| < 12 \]